Math Problem Statement

Given the following notations, write a formula for x, the break-even sales level. SP = selling price per unit FC = total fixed cost VC = variable cost per unit x =

Solution

The break-even sales level xx is the point where total revenue equals total costs. Using the given notations:

  • SPSP = selling price per unit,
  • FCFC = total fixed cost,
  • VCVC = variable cost per unit,
  • xx = number of units sold (break-even point).

Formula for the break-even point:

At the break-even point: Total Revenue=Total Costs\text{Total Revenue} = \text{Total Costs}

Total Revenue is: SP×xSP \times x

Total Costs are the sum of fixed costs and variable costs: FC+VC×xFC + VC \times x

Setting the two equal to each other: SP×x=FC+VC×xSP \times x = FC + VC \times x

Now, solve for xx: SP×xVC×x=FCSP \times x - VC \times x = FC x(SPVC)=FCx(SP - VC) = FC x=FCSPVCx = \frac{FC}{SP - VC}

Thus, the break-even sales level is: x=FCSPVCx = \frac{FC}{SP - VC}

This is the formula for the break-even sales level.

Would you like any further explanation or details on how this formula works?

Here are 5 related questions:

  1. How do changes in the selling price affect the break-even point?
  2. What happens to the break-even point if the variable cost per unit increases?
  3. Can you calculate the break-even sales level if given specific values for SP, VC, and FC?
  4. How does the break-even formula adjust for multiple products with different prices and costs?
  5. What is the relationship between contribution margin and the break-even point?

Tip: Always ensure that the selling price is greater than the variable cost; otherwise, the business will incur a loss for each unit sold.

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Math Problem Analysis

Mathematical Concepts

Algebra
Break-even analysis
Linear equations

Formulas

Break-even formula: x = FC / (SP - VC)

Theorems

Break-even point theory

Suitable Grade Level

Grades 9-12